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This paper discusses coordinated control problems for systems over finite fields via fully actuated approach. Novel types of consensus and synchronization, which are irrelevant to initial states, are proposed. For different coordinated control problems, suitable controllers are designed, which are able to drive the system to the corresponding trajectory in finite steps. Besides, the conclusions are applied to stabilization, and the convergence time is analyzed. As an application, a camera network example is given to illustrate the results.

This paper mainly studied the problem of solving interval type-2 fuzzy relation equations $ \widetilde A \circ \widetilde X = \widetilde B $. First, to solve the interval type-2 fuzzy relation equations, we extend the semi-tensor product of matrices to interval matrices and give its specific definition. Second, the interval type-2 fuzzy relation equation was divided into two parts: primary fuzzy matrix equation $ {\widetilde A_\mu } \circ {\widetilde X_\mu }{\rm{ = }}{\widetilde B_\mu} $ and secondary fuzzy matrix equation $ {\widetilde A_f} \circ {\widetilde X_f} = {\widetilde B_f} $. Since all elements of $ {\widetilde X_f} $ equal to one, only the principal fuzzy matrix equation needs to be considered. Furthermore, it was proved that all solutions can be obtained from the parameter set solutions if the primary fuzzy matrix equation is solvable. Finally, with semi-tensor product of interval matrices, the primary fuzzy matrix equation was transformed into an algebraic equation and the specific algorithm for solving an interval type-2 fuzzy relation equation was proposed.

Finite-field networks (FFNs) are a class of multi-agent systems over finite fields with sensing, computing, and communication capabilities. FFNs have been investigated extensively to save computing and communication resources. This paper summarizes the current research results to provide a direction for future research. First, different models of FFNs are reviewed, including FFNs with time-delays, switching topology, and leader-following structures. Then, the consensus and synchronization problems of multi-agent systems over finite fields are analyzed, and the necessary and sufficient conditions for consensus and synchronization of some autonomous systems have been derived in recent research. Finally, the distributed control of multi-agent systems over finite fields has been developed by many scholars based on various approaches.

Summary As regulatory networks that simulate and establish gene interactions, Boolean networks (BNs) exhibit significant delays in state updates or trajectory tracking due to environmental factors in practical applications. This paper provides a comprehensive knowledge overview of Boolean control networks with time delays (TBCNs) via the semi‐tensor product theory (STP), which includes the history of BNs, the property of STP, several issues of TBCNs and future prospects. The related problems contain controllability and reachability, observability and detectability, stability and stabilization, synchronization, output tracking and regulation, and other issues. Finally, it also points out that some areas still need further research and promotion, such as disturbance decoupling, system decomposition, identification, deep reinforcement learning, and so forth.

Design of distributed pinning controllers for set stabilization of $ k $-valued logical control networks is investigated in this paper. Firstly, by analyzing the coupling correlations among nodes, pinned node set is determined. Secondly, based on the solvability of a class of matrix equations, sufficient conditions which resort to local information are put forward for the design of pinning controllers. Furthermore, an algorithm for designing pinning feedback controllers is presented, where the designed controllers are related to part of state variables instead of all variables. Finally, two illustrative examples are presented to verify the effectiveness of the main results.

In this paper, the reachability of dimension-bounded linear systems is investigated. Since state dimensions of dimension-bounded linear systems vary with time, the expression of state dimension at each time is provided. A method for judging the reachability of a given vector space $ \mathcal{V}_{r} $ is proposed. In addition, this paper proves that the $ t $-step reachable subset is a linear space, and gives a computing method. The $ t $-step reachability of a given state is verified via a rank condition. Furthermore, annihilator polynomials are discussed and employed to illustrate the relationship between the invariant space and the reachable subset after the invariant time point $ t^{\ast} $. The inclusion relation between reachable subsets at times $ t^{\ast}+i $ and $ t^{\ast}+j $ is shown via an example.

The simplification problem of logical functions is investigated via the matrix method. First, necessary and sufficient conditions are put forward for the decomposition of logical matrices. Based on this, several criteria are proposed for the simplification of logical functions. Furthermore, an algorithm, which can derive simpler logical forms, is developed, and illustrative examples are given to verify the effectiveness. Finally, the obtained theoretical results are applied to the simplification of electric circuits.

The set of associative and commutative hypercomplex numbers, called the perfect hypercomplex algebras (PHAs) is investigated. Necessary and sufficient conditions for an algebra to be a PHA via semi-tensor product (STP) of matrices are reviewed. The zero sets are defined for non-invertible hypercomplex numbers in a given PHA, and characteristic functions are proposed for calculating zero sets. Then PHA of various dimensions are considered. First, classification of 2-dimensional PHAs are investigated. Second, all the 3-dimensional PHAs are obtained and the corresponding zero sets are calculated. Finally, 4- and higher dimensional PHAs are also considered.

A kind of algebra, called numerical algebra, is proposed and investigated. As its opponent, non-numerical algebra is also defined. The numeralization and dis-numeralization, which convert non-numerical algebra to numerical algebra and vise versa, are considered. Product structure matrix (PSM) of a finite dimensional algebra is constructed. Using PSM, some fundamental properties of finite dimensional algebras are obtained. Then a necessary and sufficient condition for a numerical algebra to be a field is presented. Finally, the invertibility of Segre (commutative) quaternion and some related properties of matrices over Segre quaternion are investigated.

In this article, the output tracking of Boolean control networks with impulsive effects (BCNs-IE) is discussed. Based on structure matrices of BCNs-IE and controllability matrices with respect to state subsets, stable complex attractors are studied. By constructing an auxiliary BCN-IE and using stable complex attractors, a necessary and sufficient condition for the output tracking problem is proposed. In addition, an algorithm to design state feedback controllers for the output tracking problem is provided. Moreover, an example is given to show the validity of the obtained results.

Nonlinear feedback shift registers (NFSRs) are the main components of stream ciphers and convolutional decoders. Recent years have seen an increase in the requirement for information security, which has sparked NFSR research. However, the NFSR study is very imperfect as a result of the lack of appropriate mathematical tools. Many scholars have discovered in recent years that the introduction of semi-tensor products (STP) of matrices can overcome this issue because STP can convert the NFSR into a quasi-linear form. As a result of STP, new NFSR research has emerged from a different angle. In view of this, in order to generalize the latest achievements of NFSRs based on STP and provide some directions for future development, the research results are summarized and sorted out, broadly including the modeling of NFSRs, the analysis of the structure of NFSRs, and the study of the properties of NFSRs.

Since the 1930s, topological counting on finite sets has been an interesting work so as to enumerate the number of corresponding order relations on the sets. Starting from the semi-tensor product (STP), we give the expression of the relationship between subsets of finite sets from the perspective of algebra. Firstly, using the STP of matrices, we present the algebraic representation of the subset and complement of finite sets and corresponding structure matrices. Then, we investigate respectively the relationship between the intersection and union and intersection and minus of structure matrices. Finally, we provide an algorithm to enumerate the numbers of topologies on a finite set based on the above theorems.

For discrete fuzzy descriptor systems with time-delays, the problem of designing fuzzy observers is investigated in this paper. Based on an equivalent transformation, discrete fuzzy descriptor systems with time-delays are converted into standard discrete systems with time-delays. Then, via linear matrix inequality (LMI) approach, both delay-dependent and delay-independent conditions for the existence of fuzzy state observers are obtained. Finally, two numerical examples are provided to illustrate the proposed method.

Solutions are constructed for the Kalman-Yakubovich-transpose equation X−AXTB=C. The solutions are stated as a polynomial of parameters of the matrix equation. One of the polynomial solutions is expressed by the symmetric operator matrix, controllability matrix, and observability matrix. Moreover, the explicit solution is proposed when the Kalman-Yakubovich-transpose matrix equation has a unique solution. The provided approach does not require the coefficient matrices to be in canonical form. In addition, the numerical example is given to illustrate the effectiveness of the derived method. Some applications in control theory are discussed at the end of this paper.

A new approach is presented for obtaining the solutions to Yakubovich-j-conjugate quaternion matrix equation X−AX^B=CY based on the real representation of a quaternion matrix. Compared to the existing results, there are no requirements on the coefficient matrix A. The closed form solution is established and the equivalent form of solution is given for this Yakubovich-j-conjugate quaternion matrix equation. Moreover, the existence of solution to complex conjugate matrix equation X−AX¯B=CY is also characterized and the solution is derived in an explicit form by means of real representation of a complex matrix. Actually, Yakubovich-conjugate matrix equation over complex field is a special case of Yakubovich-j-conjugate quaternion matrix equation X−AX^B=CY. Numerical example shows the effectiveness of the proposed results.

This paper considers the stable set and coloring problems of hypergraphs and presents several new results and algorithms using the semitensor product of matrices. By the definitions of an incidence matrix of a hypergraph and characteristic logical vector of a vertex subset, an equivalent algebraic condition is established for hypergraph stable sets, as well as a new algorithm, which can be used to search all the stable sets of any hypergraph. Then, the vertex coloring problem is investigated, and a necessary and sufficient condition in the form of algebraic inequalities is derived. Furthermore, with an algorithm, all the coloring schemes and minimum coloring partitions with the given q colors can be calculated for any hypergraph. Finally, one illustrative example and its application to storing problem are provided to show the effectiveness and applicability of the theoretical results.

This paper is concerned with the singular LQ problem for irregular singular systems with persistent disturbances. The full information feedback control method is employed to achieve the optimal control. By restricted system equivalence transformation, the system state is decomposed into free state and restricted state and the input is decomposed into free input and forced input. Some sufficient conditions for the unique existence of optimal control-state pair are derived and these conditions are all described unitedly with matrix rank equalities. The optimal control-state pair can be explicitly formulated via solving an algebraic Riccati equation and a Sylvester equation. Moreover, under the optimal control-state pair, the resulting system has no free state.